TBG II: Stable Symmetry Anomaly in Twisted Bilayer Graphene
Zhi-Da Song, Biao Lian, Nicolas Regnault, B. Andrei Bernevig
TTBG II: Stable Symmetry Anomaly in Twisted Bilayer Graphene
Zhi-Da Song, Biao Lian, Nicolas Regnault,
1, 2 and B. Andrei Bernevig ∗ Department of Physics, Princeton University, Princeton, New Jersey 08544, USA Laboratoire de Physique de l’Ecole normale superieure,ENS, Universit´e PSL, CNRS, Sorbonne Universit´e,Universit´e Paris-Diderot, Sorbonne Paris Cit´e, Paris, France (Dated: October 28, 2020)We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just thetwo active bands) with particle-hole symmetry is anomalous and hence incompatible with latticemodels. Previous works, e.g. , [Phys. Rev. Lett. 123, 036401], [Phys. Rev. X 9, 021013], [Phys.Rev. B 99, 195455], and others [1–4] found that the two flat bands in TBG possess a fragile topologyprotected by the C z T symmetry. [Phys. Rev. Lett. 123, 036401] also pointed out an approximateparticle-hole symmetry ( P ) in the continuous model of TBG. In this work, we numerically confirmthat P is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a P -protected stable topology. This stable topology implies 4 l +2 ( l ∈ N ) Diracpoints between the middle two bands. The P -protected stable topology is robust against arbitrarygap closings between the middle two bands the other bands. We further show that, remarkably, this P -protected stable topology, as well as the corresponding 4 l + 2 Dirac points, cannot be realized inlattice models that preserve both C z T and P symmetries. In other words, the continuous modelof TBG is anomalous and cannot be realized on lattices. Two other topology related topics, withconsequences for the interacting TBG problem, i.e. , the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed. I. Introduction
TBG at the first magic angle ( θ ≈ . ◦ ) exhibits agroup of two almost exactly flat bands [5]. Due to theinteresting interaction insulating and conducting states[6–36], superconductor states [37–55], and single-particletopology [1–3, 23, 56–66] in the flat bands, TBG repre-sents one of the most versatile physical systems of re-cent years [1–21, 23–35, 37–64, 66–113]. Refs. [56, 58]showed that the C z T symmetry of TBG protects a frag-ile topology [80, 114–118] of the two flat bands, which ischaracterized by a Z -valued winding number. The frag-ile topology manifests itself as a topological obstructionfor exponentially decaying Wannier functions satisfying C z T symmetry for the two flat bands. However, theWannier obstruction can be removed by adding trivialbands into the consideration [80, 114, 115]. For example,Ref. [57] showed explicitly that symmetric Wannier func-tions can be constructed if certain additional orbitals arecoupled the fragile topological band protected by C z T .However, the papers arguing for a trivialization of thebands [56, 57] neglected one (approximate) symmetry ofthe TBG model [5].The Bistritzer MacDonald (BM) model [5] of TBG hasan approximate particle-hole symmetry P first pointedout in Ref. [58]. It was already pointed out in Ref. [58]that with this approximate symmetry, there seems to bea further, stable topology in TBG, but this result wasnot further expanded. We here numerically confirm thatthe error - on the wavefunctions - of the P symmetry ∗ [email protected] (defined in Section II B) in the BM model of TBG is ex-tremely small ( < . P symmetry as agood approximation for the low energy physics in TBG.We prove that if the C z T protected winding number ofthe two flat bands is odd (true in TBG), then the twoflat bands have a stable topology protected by P , which ischaracterized by a Z invariant δ . In contrast to the frag-ile topological bands, which can be trivialized by beingcoupled to certain trivial bands, the Z topology, as wellas the Wannier obstruction implied by the Z invariant,is stable against adding trivial bands that preserve the P symmetry. We further proved that, in the presenceof C z T and P , the Z invariant δ of 2 M particle-holesymmetric bands (cid:15) − M ( k ) · · · (cid:15) − ( k ) , (cid:15) ( k ) · · · (cid:15) M ( k ) is re-lated to the number of Dirac points N D between (cid:15) − ( k )and (cid:15) ( k ) in the first Brillouin zone (BZ) as δ = N D mod2, provided that the 2 M bands are gapped from higherand lower bands. Here (cid:15) n ( k ) ( (cid:15) − n ( k )) is the n -th posi-tive (negative) band. Therefore, as long as 4 l + 2 ( l ∈ N )Dirac points exist between (cid:15) − ( k ) and (cid:15) ( k ) we find that2 M, ∀ M ∈ N particle-hole symmetric bands (separate inenergy from the M +1 , M +2 . . . and . . . , − M − , − M − δ = 1 and hence are topologically nontrivial.The feature of TBG that arbitrary 2 M bands are topo-logical is inconsistent with lattice models with C z T and P symmetry. In a lattice model, if 2 M is large enough, e.g. , equals to the number of orbitals in the model, the2 M bands have to be topologically trivial because theyspan the Hilbert space of the local orbitals. Therefore,the Z topology, and the 4 l + 2 Dirac points accordingly,cannot be realized in lattice models with finite number oforbitals. We hence call the Z topology an anomaly of the C z T and P symmetries. We further note that this im-plies that the many-body U(4) and U(4) × U(4) symme- a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t tries [110–112] are incompatible with a lattice model andhence anomalous. It also implies that the lattice modelsbuild to model TBG [1, 2, 23, 57] have to break the P symmetry or the C z T symmetry of the TBG model.This paper is organized as follows. In Section II, wepresent a review of the BM model of TBG and sum-marize its symmetries. The error of the approximateparticle-hole symmetry P is defined and is confirmed asbeing small ( < . P symmetry protects a stable Z topological state. In Sec-tion IV, a no-go theorem of the Z topology is proved forlattice models with the C z T and P symmetries. The re-lation between the Z invariant and the number of Diracpoints is also established in this section. In Section V, weshow that, when the two flat bands are gapped from theother bands, there is natural choice of Chern band ba-sis (with opposite Chern numbers) in the two flat bands.The Chern band basis is used in our interacting works[110–113] on TBG. In Section VI, we show that, in theso-called second chiral limit, defined in [110] as the sec-ond limit having an interacting extended U(4) × U(4)symmetry, the symmetry anomaly of TBG manifests asa perfect metal phase, where all the bands are connectedto each other. A brief summary of this work is given inSection VII.
II. The BM model of twisted bilayer graphene andits symmetries
We first present a short review of the BM model. Amore detailed account can be found in supplementarymaterial of Ref. [58].
A. A brief review of the BM model
TBG is an engineered material of two graphene layerstwisted by a small angle θ from each other. The bandstructure of each of the two layers exhibits two Diracpoints at the K and K (cid:48) momenta in the single layer Bril-louin zone (BZ), respectively; the two Dirac points arerelated by time-reversal T . Thus the band structure ofTBG exhibits four Dirac points: two from the top layerand the other two from the bottom layer. When θ is smallsuch that the interlayer coupling is smooth in real space -with a length scale much larger than the atom distances- the graphene valley ( K and K (cid:48) ) is a good quantumnumber of low energy states of TBG [5]. In this case, thestates around K ( K (cid:48) ) in the top layer only couple to thestates around K ( K (cid:48) ) in the bottom layer. Therefore, thelow energy band structure of TBG decomposes into twoindependent graphene valleys, and each valley has twoDirac points originated from the two layers, respectively.In this work, we will focus on the valley K . The bands inthe other valley K (cid:48) can be obtained by acting T on thebands in the valley K .We assume the top single graphene layer is rotated from the x -direction by an angle θ (rotation axis is z ).Thus the Dirac Hamiltonian around K in the top layer is − iv F ∂ x (cos θ σ x − sin θ σ y ) − iv F ∂ y (cos θ σ y + sin θ σ x ) ≈− iv F ∂ r · σ + i θ v F ∂ r × σ , where v F is the Fermi-velocity ofsingle-layer graphene and σ = ( σ x , σ y ) are Pauli matricesrepresenting the A/B sublattices of graphene. The bot-tom layer is rotated from the x -direction by an angle − θ .Correspondingly, the Dirac Hamiltonian around K in thebottom layer is − iv F ∂ r · σ − iθv F ∂ r × σ . The interlayercoupling is encoded in a position dependent matrix T ( r ),where r = ( x, y ), such that the Hamiltonian of TBG, tolinear order of θ , can be written as H ( r ) = − iv F (cid:18) τ ∂ r · σ − θ τ z ∂ r × σ (cid:19) + (cid:18) T ( r ) T † ( r ) 0 (cid:19) . (1)Here τ and τ z are the two-by-two identity matrix andthe third Pauli matrix for the layer degree of freedom,respectively. According to Ref. [5], when θ is small ( ∼ ◦ ), T ( r ) forms a smooth moir´e potential: T ( r ) = (cid:88) i =1 e − i q i · r T i , (2)where q i ’s are q = k D (0 , − q = k D ( √ , ), q = k D ( − √ , ), with k D = 2 | K | sin θ being the distance be-tween K momenta in the two layers, and T i ’s are T i = w σ + w (cid:104) σ x cos 2 π ( i − σ y sin 2 π ( i − (cid:105) , (3)where w and w are two constant parameters. Since the w term contributes to the diagonal elements, it repre-sents the interlayer coupling between the A(B) sublatticeof the top layer and the A(B) sublattice of the bottomlayer. Similarly, the w term only contributes to the off-diagonal elements, it is thus associated to the interlayercoupling between A(B) sublattice of the top layer andB(A) sublattice of the bottom layer.The moir´e potential (Eq. (2)) is invariant (up to agauge transformation) under the translations a M = πk D ( √ , ), a M = πk D ( − √ , ). The translation sym-metry of the moir´e potential is manifest in real space(Fig. 1a). The corresponding reciprocal lattice bases are b M = q − q , b M = q − q (Fig. 1b). The unit cellspanned by a M and a M is referred to as the moir´e unitcell. Each moir´e unit cell has one AA region, one ABregion, and one BA region. In the AA region, the A(B)sublattice of the top layer sit on top of the A(B) sublat-tice of the bottom layer; in the AB region, the A and Bsublattices of the top layer sit on top of the B sublatticesand the empty hexagon centers of the bottom layer, re-spectively; in the BA region, the B and A sublattices ofthe upper layer sit on top of the A sublattices and theempty hexagon centers of the lower layer, respectively.First principle calculations show that the two layers arecorrugated in the z -direction [119–122]. The distance be-tween the two layers in the AA region is larger than the K M � � M M K M K M � � M M K M B a nd E n e r gy ( m e V ) B a nd E n e r gy ( m e V ) W il s on L oop B a nd s ( � � ��� ��� k � k � a M1 a M2 AA AB BA K M M M � � q q q (a) (d) (e)(f) (g) b M1 b M2 � ������������� W il s on L oop B a nd s ( � � k x k y (c) Lattice model2 Dirac fermions with C 𝒯 and 𝒫 symmetries incompatible (b) q q q KK'
FIG. 1. The lattice, symmetry anomaly, band structures, and Wilson loop bands of TBG. (a)
The moir´e unit cell, where theblue sheet and the red sheet represent the top and bottom layers, respectively. In the AA, AB, BA regions, the A sublattice ofthe top layer are located above the A sublattice, the B sublattice, and the hexagon center of the bottom layer, respectively. (b)
The moir´e Brillouin zone. Left: The grey and yellow hexagons represent the moir´e Brillouin zone for the graphene valleys K and K (cid:48) , respectively. Right: The reciprocal lattices and the high symmetry momenta of the moir´e Brillouin zone in graphenevalley K . (c) l + 2 ( l ∈ N ) Dirac points cannot be realized in lattice models with C z T and P symmetries. (d-e) The bandstructure and Wilson loop bands of the middle two bands (shaded) at θ = 1 . ◦ . The crossings in Wilson loop bands areprotected by C z T and/or the approximate P . Each Wilson loop operator is integrated along b M and the spectrum is plottedalong b M . (f-g) The band structure and Wilson loop bands of the middle ten bands (shaded) at θ = 0 . ◦ . The crossings at λ = 0 , π in the Wilson loop bands are protected by C z T and/or by the approximate P ; the double degeneracies with λ (cid:54) = 0 , π at k = 0 , π are protected by the approximate P . These double-degeneracies guarantee Wilson loop flow for any bands with4 n + 2 Dirac nodes at zero energy. In fact, we have kept the P -breaking term iθv F τ z ∂ r × σ (Eq. (1)) in the calculations usedto generate this plot, which would split the double degeneracies in principle. However, the splittings are almost invisible byeye in the plot, implying that the P symmetry is a good approximation. The degeneracies are exact when P is exact. Theparameters of Hamiltonian used in (d-g) are v F = 5 . · ˚A, | K | = 1 . − , w = 110meV, w = 0 . w . distance in the AB and BA regions. Since w and w are mainly dominated by the couplings in the AA andAB/BA regions [5], respectively, this implies that, in therealistic model, w is smaller than w [2]. In Figs. 1dand 1f, we show the the band structures for two differ-ent twist angles θ = 1 . ◦ , . ◦ . The parameters are setas v F = 5 . · ˚A, | K | = 1 . − , w = 110meV, w = 0 . w . B. Symmetries of the BM model
The model Eq. (1) has several point group symme-tries: (i) C z T = σ x K , where K is the complex conjuga-tion, (ii) C z = e i π σ z , (iii) C x = τ x σ x . One can verifythat the Hamiltonian is invariant under these symme-tries. Notice that the single-graphene-valley Hamiltoniandoes not have the C z rotation and the time-reversal T symmetries since they map one graphene valley to theother. The three crystalline symmetries and the moir´etranslations generate the magnetic space group P (cid:48) (cid:48) P = iτ y ,which transforms the position as r → − r [58]. Here τ x,y,z are Pauli matrices representing the layer degree of free-dom. Under P the Hamiltonian transforms as P H ( r ) P † = − H ( − r ) + iθv F τ z ∂ r × σ . (4)When θ ∼ ◦ , the second term on the right hand sideis of order 0 . v F k D and hence is much smaller thanthe energy scale of the low energy physics, which is oforder v F k D . Therefore, P is an emergent anticommutingsymmetry when θ is small. It satisfies the algebra [58]:[ C z T, P ] = [ C z , P ] = 0 , { C x , P } = 0 , P = − . (5)For later convenience, we define an anti-unitary particleoperation P = P C z T = iτ y σ x K , which is local in realspace and satisfies P = −
1. It acts on the Hamiltonianas P H ( r ) P − = − H ( r ) + iθv F τ z ∂ r × σ . (6)We can also define a chiral operation C = σ z , whichis local in real space. [59]. Under C the Hamiltoniantransforms as CH ( r ) C † = − H ( r ) + 2 w τ x σ (cid:88) i =1 e − i q i · r . (7) � =1.05 ∘ � =1.5 ∘ � =2 ∘ e rr o r( 𝒫 ) e rr o r( C ) w / w w / w -3 Zoom in (a) (b) � =1.05 ∘ � =1.5 ∘ � =2 ∘ FIG. 2. Errors of the approximate symmetries P (a) and C (b) on the wavefunctions (as defined in Eqs. 9 and 10) in TBGas functions of w /w . Here we change w while keeping w fixed (110meV). The errors are shown for different values ofthe twist angle θ = 1 . ◦ , . ◦ , ◦ . In the so-called chiral limit [59], i.e. , w = 0, the secondterm on the right hand of side vanishes and hence C be-come an emergent anticommuting symmetry. The chiralsymmetry satisfies the algebra { C z T, C } = { C x , C } = 0 , [ C z , C ] = [ P, C ] = 0 , C = 1 . (8)We numerically checked how much P and C are brokenin the wavefunctions of the model Eq. (1). To be specific,we define the errors of the two symmetries in the two flatbands aserror( P ) = 1 − ˆ d k (2 π ) |(cid:104) u , − k |P| u − , k (cid:105)| , (9)error( C ) = 1 − ˆ d k (2 π ) |(cid:104) u , k | C | u − , k (cid:105)| , (10)respectively, where | u − , k (cid:105) and | u , k (cid:105) are the periodicparts of the Bloch states of the highest occupied bandand the lowest empty band at charge neutrality, respec-tively. When the two symmetries are exact, we have |(cid:104) u , − k |P| u − , k (cid:105)| = |(cid:104) u , k | S | u − , k (cid:105)| = 1 and hence theerrors are zero. Using the parameters v F = 5 . · ˚A, | K | = 1 .
703 ˚ A − , w = 110meV, we plot error( P ) anderror( C ) as functions of w /w (with fixed w ) for a fewtwist angles in Fig. 2. For θ = 1 . ◦ , error( P ) is small( < .
01) for w ≤ . w , thus the P symmetry is agood approximation for TBG, while the C symmetry onlystarts being good (with error < .
01) for w ≤ . w . III. Stable topology protected by particle-holesymmetry P A. The Wilson loop Z invariant protected by P We denote the Hamiltonian in momentum space as H ( k ). We assume the emergent anti-unitary particle-hole symmetry, i.e. , P H ( k ) P − = − H ( − k ), and P = − P = P C z T is anti-unitaryand squares to -1, and is the product of the unitary P ofRef. [58] and C z T . We denote the energy and the peri-odic part of Bloch state of the n -th band above (below)the zero energy as (cid:15) n ( k ) ( (cid:15) − n ( k )) and | u n ( k ) (cid:105) ( | u − n ( k ) (cid:105) ),respectively. As explained in Appendix A and in Ref.[58], | u n ( k ) (cid:105) satisfies the periodicity | u n ( k + G ) (cid:105) = V G | u n ( k ) (cid:105) , with G being a reciprocal lattice and V G a unitary matrix referred to as the embedding matrix.Since P anti-commutes with the Hamiltonian and flipsthe momentum, we have (cid:15) n ( k ) = − (cid:15) − n ( − k ). The state P| u n ( k ) (cid:105) must have the momentum − k and the energy (cid:15) − n ( − k ). In general, P| u n ( k ) (cid:105) is spanned by Bloch statesat − k as P| u n ( k ) (cid:105) = (cid:88) n (cid:48) | u n (cid:48) ( − k ) (cid:105) B ( P ) n (cid:48) n ( k ) , (11)where the summation over n (cid:48) is limited to those satisfy-ing (cid:15) n (cid:48) ( k ) = − (cid:15) n ( − k ), and B ( P ) n (cid:48) n ( k ) is a unitary matrixreferred to as the sewing matrix of P . B ( P ) ( k ) is peri-odic in momentum space, i.e. , B ( P ) ( k + G ) = B ( P ) ( k )[124, 125]. Since P = −
1, it should satisfy B ( P ) ( − k ) B ( P ) ∗ ( k ) = − . (12)Multiplying B ( P ) T ( k ) on the right hand side of the aboveequation, we obtain B ( P ) ( − k ) = − B ( P ) T ( k ) . (13)We now prove that the P symmetry protects a Z in-variant for 2 M particle-hole symmetric separate bands, i.e. , bands (cid:15) − M ( k ) , (cid:15) − M +1 ( k ) · · · (cid:15) M ( k ), gapped fromhigher and lower bands. This proof is not limitedto TBG but applies to any system having our anti-unitary P symmetry. We introduce the matrix U ( k ) =( | u − M ( k ) (cid:105) , | u − M +1 ( k ) (cid:105) · · · | u M ( k ) (cid:105) ). We parameterize k as k b + k b , where b and b are the reciprocal latticebasis vectors. Then we define the Wilson loop operatorof the 2 M bands for a given k as W ( k ) = lim N →∞ N − (cid:89) j =0 U † ( k , j πN ) U ( k , ( j + 1) 2 πN ) . (14)The order of the matrices in the product is given by j : matrices with larger k (= j πN ) always appear onthe right hand side of matrices with smaller k . Dueto the periodicity of Bloch states, W ( k ) is periodic..Since W ( k ) is unitary, its eigenvalues are phase factors e iλ n ( k x ) ( n = 1 · · · M ), where λ n ( k ) ranges from − π to π . { λ n ( k ) } are called as the Wilson loop bands. Topol-ogy is usually a result of Wilson loop flow, which in turnis a result of unavoidable crossings between Wilson loopbands.We now prove that the Wilson loop bands are doublydegenerate at k = 0 and k = π , as shown in Fig. 3a-c. In fact, we should heuristically expect this, since the � � k � 𝒫 -protected �� � k � � k � � � k ��� � k � � k � C2zT- protected W il s on l oop b a nd s ( � ) W il s on l oop b a nd s ( � ) (a) ++ ≈≈ (Kramers pair) (b) (c)(d) (e) (f) FIG. 3. Comparison of Wilson loop windings protected by P and C z T . (a-c) The Wilson loop bands with P . Thecrossings at k = 0 , π are Kramers pairs protected by P . The Z invariant (a) equals to 1 if the Wilson loop bands form azigzag connection between k = 0 and k = π and (b) equalsto 0 otherwise. The P -protected topology is stable againstadding trivial bands: Coupling the nontrivial Wilson loopbands (a) to a trivial Wilson loop bands (b) yields a nontrivialWilson loop bands (c). (d-f ) The Wilson loop bands with C z T . The crossings at λ = 0 , π are protected by C z T . Fora two-band system, the topology is nontrivial if the Wilsonloop bands winds (d) and is trivial otherwise (e). The C z T -protected topology is fragile: Coupling the nontrivial Wilsonloop bands (d) to a trivial Wilson loop bands (f) yields atrivial Wilson loop bands (f). Wilson loop respects P as it contains all bands relatedby the particule-hole symmetry. Since P is anti-unitaryand squares to − (cid:104) u n ( k ) | u n (cid:48) ( k (cid:48) ) (cid:105) = (cid:104)P u n (cid:48) ( k (cid:48) ) |P u n ( k ) (cid:105) = (cid:88) mm (cid:48) B ( P ) ∗ m (cid:48) n (cid:48) ( k (cid:48) ) (cid:104) u m (cid:48) ( − k (cid:48) ) | u m ( − k ) (cid:105) B ( P ) mn ( k ) . (15)In the above equation we have made use of a property ofanti-unitary symmetries: for any two states | φ (cid:105) , | ψ (cid:105) andan arbitrary anti-unitary operator O , we have (cid:104) φ | ψ (cid:105) = (cid:104)O ψ |O φ (cid:105) . Substituting this relation into Eq. (14) andusing the periodicity relations B ( P ) ( k + G ) = B ( P ) ( k )and | u n ( k + G ) (cid:105) = V G | u n ( k ) (cid:105) , we obtain W ( k ) = B ( P ) T ( k , W T ( − k ) B ( P ) ∗ ( k , . (16)Since W ( k ) is periodic at k , W ( k ) with k = 0 , π areinvariant under the particle-hole operation: W ( k ) = B ( P ) T ( k , W T ( k ) B ( P ) ∗ ( k , , ( k = 0 , π ) . (17)It is this invariance that protects degeneracies of Wilsonloop bands at k = 0 , π . To see this, we parameter-ize the unitary matrix W ( k ) as e i H ( k ) with H ( k ) be-ing a hermitian matrix periodic in k , called the WilsonHamiltonian. The eigenvalues of H ( k ) form the Wilson loop bands. We can define the particle-hole operator for H ( k ) as ˜ P ( k ) = B ( P ) ( k , K such that Eq. (16) can bewritten as H ( k ) = ˜ P ( − k ) H ( − k ) ˜ P − ( − k ). We have˜ P ( k ) ˜ P ( − k ) = − H ( k ) which anti-commuteswith P , H ( k ) commutes with ˜ P . Because ˜ P ( k ) = − P , H ( k )] = 0 for k = 0 , π , the Wilson loop bands -the eigenstates of the Wilson Hamiltonian - at k = 0 , π form doublets due to the Kramers theorem.The Z invariant δ is defined such that δ = 1 if theWilson loop bands form a zigzag flow between k = 0and k = π - equivalent to a Quantum Spin Hall flow ofKramers paired Wannier centers, and δ = 0 otherwise.Examples of δ = 1 and δ = 0 with only P symmetry areshown in Figs. 3a and 3b, respectively. Fig. 3a does notcontain the C z T symmetry and is meant to depict thepossible cases with only our anti-unitary P symmetry.Because the degeneracies at k = 0 , π are protected by P ,a zigzag flow is stable against adding P -preserving bandsas long as these bands are topologically trivial (they donot exhibit Wilson loop flow themselves) that do not closethe gaps between the 2 M bands and the higher/lowerbands (Fig. 3a-c).In Figs. 1e and 1g, we plot the Wilson loop bands ofthe middle two bands ( (cid:15) − ( k ) , (cid:15) ( k )) of TBG with θ =1 . ◦ and the Wilson loop bands of the middle ten bands( (cid:15) − ( k ) · · · (cid:15) ( k )) of TBG with θ = 0 . ◦ , respectively.Both have the zigzag flow and hence have δ = 1. We donot plot the Wilson loop bands of the middle ten bands ofTBG with θ = 1 . ◦ because they have touching pointswith higher/lower bands at generic momenta (away fromhigh symmetry lines). B. Comparison of the P -protected topology and C z T -protected topology In Ref. [58], some of the authors of the present workproved that the C z T symmetry protects the Wilson loopflow for two bands, as shown in Fig. 3d, where the cross-ings at λ = 0 , π are protected by C z T . The Wilsonloop flow is characterized by an integer-valued invariant e : the winding number of a smooth branch of the Wil-son loop bands. There is a gauge ambiguity for the signof e . For example, the Wilson loop bands in Fig. 3dhas e = 1 if we choose the branch going up to define thewinding number and e = − e is also referred to as the Euler’s class [56], aswill be briefly introduced in Section V. With only C z T symmetry, the flow can be broken by adding two trivial(flat) Wilson loop bands, as shown in Fig. 3d-f, since thecrossings at generic positions - different from λ = 0 , π -in the Wilson loop spectrum are not protected by C z T .After the Wilson loop bands are gapped, one can stilldefine a C z T -protected Z invariant through the nestedWilson loop [56, 58]. Nevertheless, this C z T -protected Z invariant does not correspond to Wannier obstruc-tion [56, 57]. Therefore, the topology protected only by C z T is fragile. Ref. [58] showed that, by adding theunitary particle-hole symmetry P , one cannot render theStiefel–Whitney class [56] trivial by adding more bands;however, nontrivial Stiefel-Whitney index does not implynon-Wannierizable bands, and hence [58] called the index“stable”, between quotation marks; this paper removesthe quotation marks by proving non-wannieralizability.On the contrary, with the P symmetry, we cannotbreak the zigzag flow by adding trivial (non-winding)Wilson loop bands, just like in the Quantum Spin Hallproblem. First, due to the Kramers degeneracy guaran-teed by P , a trivial state must have at least two Wil-son loop bands - corresponding to the fact that, withparticle-hole symmetry, we must add to the nontrivialbands, generically, two bands - of some energy ± E . Thetwo Wilson loop bands are separated at generic k butdegenerate at k = 0 , π , as shown in Fig. 3b. If we couplesuch a two-band trivial state to the topological state, thetotal Wilson loop bands are still gapless (Fig. 3c) sincethe degeneracies at k = 0 , π are protected. Therefore,the topology protected by P is stable.If a two-band system has both C z T and P symme-tries, the Z invariant protected by P is given by theparity of e , i.e. , δ = e mod 2. For example, the Wilsonloop bands in Fig. 1e has e = 1 and δ = 1. There isstable topology from P , in systems with an even numberof bands. TBG has even number of bands (as it has to,since P = C z T P implies even number of bands: nonzeroenergy E (cid:54) = 0 states come in pairs ± E , while zero energystates have Kramers degeneracy since P = − l + 2 ( l ∈ N ) Dirac nodesat zero energy (proved in Section IV), which will showthat TBG is in the topologically nontrivial class of thissymmetry. C. An alternative expression of the Z invariant We have mentioned that the zigzag flow of the Wil-son loop bands protected by P is same as the zigzagflow of Wilson loop bands protected by the time-reversalsymmetry in 2D Quantum Spin Hall topological insula-tor [126]. Now we show that they are indeed equiva-lent. Suppose H ( k ) have the P ( P = −
1) symmetry, i.e. , H ( k ) = −P H ( − k ) P − , then we define the squaredHamiltonian as H ( k ) = H ( k ) · H ( k ) such that it com-mutes with P , i.e. , H ( k ) = P H ( − k ) P − . We canregard P as a “time-reversal symmetry” of H ( k ). Aneigenstate of H ( k ) with the energy (cid:15) n ( k ) is still an eigen-state of H ( k ) but has the squared energy (cid:15) n ( k ). Statesof the 2 M particle-hole-symmetric bands used to definethe Wilson loop (Eq. (14)), i.e. , | u − M ( k ) (cid:105) · · · | u M ( k ) (cid:105) ,form the lowest 2 M bands of the squared Hamiltonian H ( k ). Thus the Wilson loop operator of the 2 M particle-hole symmetric bands of H ( k ) is same as theWilson loop operator of the 2 M lowest bands of H ( k ).The zigzag flow of the Wilson loop can be equivalentlythought as protected by the “time-reversal symmetry” of � k ����� ∂ ℬℬ FIG. 4. The path used to define the Z invariant protectedby P . B = [ − π, π ] ⊗ [ − π,
0] is half of the Brillouin zone. Itsboundary ∂ B is invariant under the particle-hole symmetry P ( k → − k ). With the C z T symmetry, one Dirac point in B contributes to a π Berry’s phase along ∂ B . H ( k ).The time-reversal-protected Z invariant can be alter-natively expressed as a topological obstruction [128, 129].Consider 2 M bands {| u In ( k ) (cid:105) , | u IIn ( k ) (cid:105) | n = 1 · · · M } ina time-reversal ( T ) symmetric system that satisfy thegauge condition | u IIn ( − k ) (cid:105) = T | u In ( k ) (cid:105) , | u In ( − k ) (cid:105) = −T | u IIn ( k ) (cid:105) , then the corresponding Z invariant is givenby δ = 12 π (cid:18) ‰ ∂ B d k · A ( k ) − ˆ B d k Ω( k ) (cid:19) mod 2 , (18)where B is half of the BZ whose boundary ∂ B is P -invariant, A ( k ) = i M (cid:88) n =1 (cid:88) a = I,II (cid:104) u an ( k ) | ∂ k u an ( k ) (cid:105) (19)is the Berry’s connection of the considered bands, andΩ( k ) = ∂ k × A ( k ) is the Berry’s curvature. An exampleof B is shown in Fig. 4. We regard | u − M ( k ) (cid:105) · · · | u M ( k ) (cid:105) as the lowest 2 M bands of H ( k ) and P the “time-reversal symmetry” of H ( k ). If we impose the gauge | u − n ( − k ) (cid:105) = P| u n ( k ) (cid:105) ( n = 1 · · · M ), i.e. , choosethe sewing matrix defined in Eq. (11) as B ( P ) n (cid:48) ,n ( k ) = δ n (cid:48) , − n sgn( n ), then we can regard | u n ( k ) (cid:105) and | u − n ( k ) (cid:105) as | u In ( k ) (cid:105) and | u IIn ( k ) (cid:105) , respectively. Thus the Z invari-ant of the 2 M bands of H ( k ) protected by P is given byEq. (18). This expression will be used for one of the waysto prove the symmetry anomaly of 4 l + 2 Dirac points insystems with C z T and P symmetries (See Section IV). IV. A no-go theorem of two Dirac fermions onlattices with C z T and P symmetries In this section, we will prove that if there are 4 l + 2( l ∈ N ) Dirac fermions at zero energy (chemical poten-tial) in a system with C z T and P symmetries, then the Z invariant of the 2 M bands (arbitrary M ) above andbelow the chemical potential, i.e. , (cid:15) − M ( k ) · · · (cid:15) M ( k ), isguaranteed to be 1, provided that the 2 M bands aregapped from other bands. As a consequence, for arbi-trary M , the 2 M particle-hole symmetric bands are notWannierizable. That means the 4 l + 2 Dirac fermions donot have a lattice support.Before going into a mathematical proof, we first givean intuitive proof that the Wilson loop of a C z T and P system with 4 l + 2 Dirac fermions at zero energy needsto wind. We first assume that we have 2 bands sep-arate from other bands close to charge neutrality. In[48, 56] it was shown that the number of Dirac points inhalf BZ mod 2 equals the winding of the Wilson loop.For 4 l + 2 Dirac nodes in between these two bands, thewinding would be odd, as in Fig. 1e. Adding non-zerotrivial or nontrivial energy bands to this system wouldhappen in pairs; introducing a set (trivial, due to C z T ,which renders Chern numbers to be zero and hence makesany single band topologically trivial) bands at non-zeroenergy would have its P conjugate and appear in num-bers 2 n . Introducing nontrivial bands at nonzero energywould mean introducing 2 × n bands into the system,as any possible nontrivial set of bands at a given energycomes as a multiple of 2. These bands can introduce onlya multiple of 4 number of Dirac fermions into the system:each set of two separate bands has to have a multiple of2 Dirac fermions. From our Quantum Spin Hall (QSH)experience, whatever number of bands we introduce ontop of our nontrivial bands with 4 l + 2 Dirac fermionscannot change the Wilson loop winding, as we are ei-ther adding trivial bands or pairs of nontrivial bands toa QSH system. Hence the winding (of the 4 n + 2 Diracfermion band) is stable to the addition of any bands re-specting C z T and P . The only way the winding can beinterrupted is by the addition of one set of 2 bands withWilson loop winding to the already existent Wilson loopwinding 2-bands. However, since with C z T the numberof Dirac nodes mod 4 is equal to twice times the wind-ing, this additional one set of 2-bands would bring aboutanother 4 l (cid:48) +2 Dirac points so the full system would havea number of Dirac fermions divisible by 4. Hence a sys-tem with 4 l + 2 Dirac fermions and C z T and P has toexhibit Wilson loop winding.Now, by making use of Eq. (18), we give another proofthat 2 M bands (gapped from other bands) with C z T and P symmetries that have 4 l + 2 Dirac points between (cid:15) − ( k ) and (cid:15) ( k ) must have a nontrivial topology. Dueto the C z T symmetry, the Bloch states satisfy C z T | u n ( k ) (cid:105) = (cid:88) n (cid:48) | u n (cid:48) ( k ) (cid:105) B ( C z T ) n (cid:48) n ( k ) , (20)where B ( C z T ) n (cid:48) n ( k ) is unitary and called the C z T sewingmatrix. The summation over n (cid:48) is limited to values sat-isfying (cid:15) n (cid:48) ( k ) = (cid:15) n ( k ). Substituting this constraint intothe definition of the Berry’s curvature Ω( k ), we find thatΩ( k ) = 0 [48, 56, 80]. Thus we only need to evaluate thefirst term on the right hand side of Eq. (18). We define A (cid:48) ( k ) = i (cid:80) Mn =1 (cid:104) u n ( k ) | ∂ k u n ( k ) (cid:105) for the positive bands and A (cid:48)(cid:48) ( k ) = i (cid:80) Mn =1 (cid:104) u − n ( k ) | ∂ k u − n ( k ) (cid:105) for the negativebands. The total Berry’s connection is A = A (cid:48) + A (cid:48)(cid:48) .By imposing the gauge condition | u − n ( − k ) (cid:105) = P| u n ( k ) (cid:105) ( n = 1 · · · M ) required by Eq. (18), we find A (cid:48)(cid:48) ( k ) = i M (cid:88) n =1 (cid:104) u − n ( k ) | ∂ k u − n ( k ) (cid:105) = i M (cid:88) n =1 (cid:104) ∂ k P u − n ( k ) |P u − n ( k ) (cid:105) = i M (cid:88) n =1 (cid:104) ∂ k u n ( − k ) | u n ( − k ) (cid:105) = − i M (cid:88) n =1 (cid:104) u n ( − k ) | ∂ k u n ( − k ) (cid:105) = A (cid:48) ( − k ) , (21) where we have applied the property of anti-unitary sym-metry introduced below Eq. (15). Since the boundary ∂ B (Fig. 4) is invariant under k → − k , the integrals of A (cid:48) ( k ) and A (cid:48)(cid:48) ( k ) are equal, i.e. , ‰ ∂ B d k · A (cid:48)(cid:48) ( k ) = ‰ ∂ B d k · A (cid:48) ( k ) . (22)The C z T symmetry stabilizes 2D Dirac points [130], andeach Dirac point between the positive bands and thenegative bands contribute to a π or − π Berry’s phaseof A (cid:48) ( k ) (Fig. 4). Due to the P symmetry (cid:15) − n ( − k ) = − (cid:15) n ( k ), the Dirac points must be equally distributed in B and its complementary set BZ - B . Hence if there are4 l + 2 Dirac points in the BZ, there will be 2 l + 1 Diracpoints in B and we have ¸ ∂ B d k · A (cid:48) ( k ) = (2 l + 1) π mod2 π . According to Eq. (22), we have ‰ ∂ B d k · ( A (cid:48) ( k ) + A (cid:48)(cid:48) ( k )) = (4 l + 2) π mod 4 π. (23)Substituting this equation into Eq. (18) and using thefact that Ω( k ) = 0, we obtain δ = 1. Thus the pres-ence of 4 l + 2 Dirac points in a system with C z T and P symmetries implies a nontrivial topology. In contrast tolattice models whose whole bands are trivial, this non-trivial topology is guaranteed by the Dirac points be-tween (cid:15) ( k ) and (cid:15) − ( k ) and hence cannot be trivializedby adding higher and lower energy bands (preserving P ).Therefore, no matter how many high energy bands areincluded, as long as they respect C z T and P , the con-sidered bands must have have nontrivial topology. Aswill be shown in next paragraph, in a lattice model witha finite number of orbitals per unit cell, the Wilson loopbands of the whole bands must be trivial. Therefore,4 l + 2 Dirac points cannot be realized in lattice modelsbecause the corresponding band structure, no matter howmany high and low energy bands are considered, must betopologically nontrivial.Here we show that the whole bands of a latticemodel must be trivial. Let the lattice model has N orbitals, then the U ( k ) matrix entering the Wil-son loop operator (Eq. (14)) of the whole bands is U ( k ) = ( | u ( k ) (cid:105) · · · | u N ( k ) (cid:105) ). By the completeness ofall the Bloch states we have U ( k ) U † ( k ) = 1. Thusthe Wilson loop operator in Eq. (14) is W ( k ) = U † ( k , U ( k , π ) = U † ( k , V (0 , π ) U ( k , V (0 , π ) is the embedding matrix defined in Appendix A(with G = 2 π b ). Since U ( k ,
0) is an N × N unitarymatrix, the eigenvalues of W ( k ) are same as eigenvaluesof V (0 , π ) and hence do not change with k and do notwind.It is worth noting that, in TBG, the symmetryanomaly does not depend on the parameters of theHamiltonian Eq. (1). In the weak coupling limit ( w (cid:28) v F k D , w (cid:28) v F k D ), we have two Dirac points at K M and K (cid:48) M in the moir´e BZ. If the 2 M bands (cid:15) − M ( k ) · · · (cid:15) M ( k )are gapped from the other bands, the 2 M bands mustbe topological due to correspondence between the num-ber of Dirac points and the Z invariant δ . Tuning theparameters of TBG may couple the 2 M bands to higherbands (cid:15) M +1 ( k ) · · · (cid:15) M (cid:48) ( k ) ( M (cid:48) > M ) and lower bands (cid:15) − M (cid:48) ( k ) · · · (cid:15) − M − ( k ), which are assumed be gappedfrom (cid:15) M (cid:48) +1 ( k ) and (cid:15) − M (cid:48) − ( k ) as we tune the parame-ters. In the weak coupling limit, the additional 2 M (cid:48) − M bands must have δ = 0 since they do not have Diracpoints between (cid:15) − M − ( k ) and (cid:15) M +1 ( k ). Therefore, afterwe couple the 2 M bands to the 2 M (cid:48) − M bands, the 2 M (cid:48) bands as a whole will have δ = 1 + 0 = 1. As we tune theparameters, additional Dirac points between (cid:15) ( k ) and (cid:15) − ( k ) may be created due to gap closing and reopeningbetween (cid:15) ( k ) and (cid:15) − ( k ). However, the total numberof Dirac points between (cid:15) ( k ) and (cid:15) − ( k ) must equal to2 mod 4, i.e. , 4 l + 2 ( l ∈ N ), because the topologicalinvariant of the 2 M (cid:48) bands is guaranteed to be δ = 1. V. The Chern band basis
In this section we show that, if the two bands (cid:15) − ( k )and (cid:15) ( k ) are gapped from other bands, we can recombinethem as two Chern bands with Chern numbers e and − e , with e being the Euler’s class [56, 131–133] (or,equivalently, the Wilson loop winding number protectedby C z T [48]). (In TBG, the Chern numbers given by ± e are also equal to the e Y = ± y matrix in the 2-dimensional space of n = ± e . (We refer thereaders to Refs. [56, 131, 48] for more details.) Sup-pose the two bands (cid:15) ( k ) and (cid:15) − ( k ) are gapped fromother bands. Then, at k away from Dirac points betweenthe two bands, the C z T operator leaves each band un-changed up to a phase factor. In other words, the C z T sewing matrix (Eq. (20)) is diagonal at these k . Hencein general, the C z T symmetry acts on the Bloch statesas C z T | u n ( k ) (cid:105) = | u n ( k ) (cid:105) e iθ n ( k ) ( n = ± θ n ( k )being the phase factors. According to Ref. [56], it fol-lows that the non-Abelian Berry’s connection of the two bands at k away from Dirac points takes the form A nn (cid:48) ( k ) = i (cid:104) u n ( k ) | ∂ k u n (cid:48) ( k ) (cid:105) = (cid:32) − ∂ k θ ( k ) − i a ( k ) e i θ k ) − θ − k )2 i a ( k ) e i θ − k ) − θ k )2 − ∂ k θ − ( k ) (cid:33) nn (cid:48) . (24) a ( k ) is a gauge invariant quantity up to a global ambi-guity of ± sign. The Euler’s class is given by e = 12 π (cid:88) i ‰ ∂D i d k · a ( k ) = 12 π ˆ BZ (cid:48) d k f ( k ) ∈ Z . (25) Here i indexes the Dirac points in the BZ, D i is asufficiently small region covering the i -th Dirac point,BZ (cid:48) = BZ − (cid:80) i D i , and f ( k ) = ∂ k × a ( k ).We introduce the two Chern band basis as | v ± ( k ) (cid:105) = 1 √ e i θ k )2 | u ( k ) (cid:105) ± ie i θ − k )2 | u − ( k ) (cid:105) ) . (26)There are two ambiguities in the above equation: (i)There is an ambiguity of the two branches of θ n , i.e. , θ n and θ n + π . (ii) At the Dirac points, where the twobands are degenerate, there is an ambiguity of choosing u ( k ) and u − ( k ). Replacing θ ( k ) / θ ( k ) / π or replacing θ − ( k ) / θ − ( k ) / π will interchange | v + ( k ) (cid:105) with | v − ( k ) (cid:105) . Similarly, interchanging | u ( k ) (cid:105) and | u − ( k ) (cid:105) at the Dirac points will also interchange | v + ( k ) (cid:105) with | v − ( k ) (cid:105) at the Dirac points. To solve theseambiguities, as detailed in Appendix B, we require thatthe Berry’s curvatures of | v + ( k ) (cid:105) and | v − ( k ) (cid:105) to be con-tinuous, or, equivalently,lim q → |(cid:104) v m (cid:48) ( k + q ) | v m ( k ) (cid:105)| = δ m (cid:48) m , (27)where m, m (cid:48) = ± . Using Eqs. (24) and (26), we cancalculate the non-Abelian Berry’s connection on the basis | v ± ( k ) (cid:105) at k away from the Dirac points. We obtain A (cid:48) mm (cid:48) ( k ) = i (cid:104) v m ( k ) | ∂ k v m (cid:48) ( k ) (cid:105) = (cid:18) a ( k ) 00 − a ( k ) (cid:19) mm (cid:48) , (28)and hence F (cid:48) mm (cid:48) = − [ ∂ k x − A (cid:48) x , ∂ k y − A (cid:48) y ] mm (cid:48) = (cid:18) f ( k ) 00 − f ( k ) (cid:19) mm (cid:48) , (29)for k not at the Dirac points. Therefore, if the Berry’scurvature does not diverge at Dirac points, which is trueas shown in next paragraph, the Chern numbers of thestates | v ± ( k ) (cid:105) are C ± = ± π ˆ d k f ( k ) = ± e . (30)To conclude this section, we show that in the chirallimit w = 0 the Chern band basis can be chosen as theeigenstates of the chiral symmetry C (Eq. (7)). We definethe sewing matrix of C as C | u n ( k ) (cid:105) = (cid:88) n (cid:48) | u n (cid:48) ( k ) (cid:105) B ( C ) n (cid:48) n ( k ) , (31)where the summation over n (cid:48) satisfies (cid:15) n (cid:48) ( k ) = − (cid:15) n ( k ).For the TBG Hamiltonian Eq. (1), the C z T and C op-erators are σ x K and σ z , respectively. Thus we have thealgebra C = 1 and { C, C z T } = 0 and hence[ B ( C ) ( k )] = 1 , (32) B ( C z T ) ( k ) B ( C ) ∗ ( k ) + B ( C ) ( k ) B ( C z T ) ( k ) = 0 . (33)As discussed at the beginning of this section, at k notat the Dirac points, we have B ( C z T ) n (cid:48) n ( k ) = δ n (cid:48) n e iθ n ( k ) .Then the solution of B ( C ) ( k ) is B ( C ) ( k ) = ± (cid:32) − ie i θ k ) − θ − k )2 ie i θ − k ) − θ k )2 (cid:33) . (34)The ± sign cannot be determined by solving Eqs. (32)and (33). In practice, one should evaluate Eq. (31) todetermine the ± sign for given | u ± ( k ) (cid:105) . We find thatthe Chern band basis Eq. (26) diagonalizes B ( C ) ( k ). Be-low Eq. (26) we have discussed the ambiguity of choos-ing | v ± ( k ) (cid:105) and we have imposed Eq. (27) to fix thisambiguity. This ambiguity of Eq. (26) can be alterna-tively solved by choosing | v ± ( k ) (cid:105) as the eigenstates of C with the eigenvalues ±
1, respectively. This choice auto-matically satisfies Eq. (27) since the states with differentchiral eigenvalues are orthogonal, i.e. , (cid:104) v − ( k ) | v + ( k (cid:48) ) (cid:105) = (cid:104) v − ( k ) | C † C | v + ( k (cid:48) ) (cid:105) = −(cid:104) v − ( k ) | v + ( k (cid:48) ) (cid:105) = 0 for arbi-trary k and k (cid:48) .The Chern band basis in Eq. (26) can also be equiva-lently defined through the Wilson loop method [86]. Be-sides, in the chiral limit, the Chern band basis we definedis equivalent to that defined in [23]. VI. Perfect metal phase of twisted bilayergraphene in the second chiral limit
In our article Ref. [110], we consider the opposite limitof the usual chiral limit: Instead of letting w = 0, wetake w to be zero. When w = 0, the model Eq. (1) hasanother chiral symmetry C (cid:48) = τ z σ z acting on the Hamil-tonian as C (cid:48) H ( r ) C (cid:48)† = − H ( r ). Thus we call this limit asthe second chiral limit. As discussed in Section II A, w and w are the interlayer couplings contributed mainlyby the AA and AB/BA regions, respectively; thus thesecond chiral limit can be (approximately) realized if thelayer distance in the AA region is smaller than the layerdistance in the AB and BA regions (shorter distancemeans stronger coupling). Such a configuration wouldbe different from the corrugation predicted by the firstprinciple calculations [119–122], where the distance inthe AA region is larger. Nevertheless, the second chirallimit might could potentially be engineered by puttingthe TBG on certain substrate, and it represents an inter-esting interacting limit [110]. We are mainly interested inthe novel electronic band structure of TBG in the second K M � M M M K M B a nd e n e r gy ( m e V ) �� ∘ (a) � M K M K M K M -K M -K M -K M M M (b) FIG. 5. Perfect metal phase of twisted bilayer graphene in thesecond chiral limit ( w = 0). (a) The band structure at θ =1 . ◦ with the parameters v F = 5 . · ˚A, | K | = 1 . − , w = 0, w = 77meV. (b) The Brillouin zone of the twistedbilayer graphene. The solid black lines represent the C x -axis and its conjugations under C z , the dashed black linesrepresent the effective mirror symmetry M x = C x I and itsconjugations under C z . The red dots represent Dirac pointsin the mirror lines. The blue dots represent Dirac points atgeneric momenta. chiral limit and hence we leave the material realizationof the second chiral limit for future study.We find that, in the second chiral limit, the n -th posi-tive (negative) band is always connected to the ( n +1)-thpositive (negative) band. As the first positive band andthe first negative band are connected through the Diracpoints, the whole bands are all connected, as shown inFig. 5a. The phase with all bands connected is referredto as the perfect metal [134] in trilayer systems, wherethe number of Dirac nodes is odd. In the current case,we also find this “perfect metal” in even number of Diracnode systems with the special chiral symmetry of thesecond chiral limit.The perfect metal phase is protected by C z T , P , and C (cid:48) . The new chiral symmetry C (cid:48) has a strange groupalgebra as it anticommutes with T and with P [111]. Wedefine the product of P and C (cid:48) as an effective inversionsymmetry I = P C (cid:48) = τ x σ z . It commutes with the Hamil-tonian, i.e. , H ( − r ) = IH ( r ) I † and H ( − k ) = IH ( k ) I † accordingly. The effective inversion operator satisfies thealgebra { C z T, I } = 0 , { P, I } = 0 , I = 1 . (35)We first show that C z T and I protect double degenera-cies at I -invariant momenta. (In TBG, the I -invariantmomenta are Γ M and the three equivalent M M .) Sincethe Hamiltonian at an I -invariant momentum commuteswith I , the Bloch states at this momentum must formeigenstates of I . Suppose | u (cid:105) is such an eigenstate with I eigenvalue 1, then we can show that C z T | u (cid:105) must havethe opposite I eigenvalue -1 due to the anti-commutationbetween C z T and I . Therefore | u (cid:105) and C z T | u (cid:105) form adoublet that has opposite I eigenvalues. This explainsthe double degeneracies at Γ M and M M shown in Fig. 5a.Next we prove that, for arbitrary even M , the M -thpositive band is connected to the ( M + 1)-th positive0bands through 4 l + 2 ( l ∈ N ) Dirac points. (For odd M , we know by counting - see Fig. 5 - that the M -thband is connected to the ( M + 1)-th band through thedouble degeneracies at the I -invariant momenta.) Weonly need to prove for the situation where the four bands (cid:15) M − ( k ), (cid:15) M ( k ), (cid:15) M +1 ( k ), (cid:15) M +2 ( k ) do not form four-folddegeneracies at high symmetry momenta since otherwise (cid:15) M ( k ) is already connected to (cid:15) M +1 ( k ). (As shown inFig. 5a, we also do not observe four-fold degeneraciesat high symmetry momenta.) We assume there are intotal n D Dirac points between the first M positive bands (cid:15) ( k ) · · · (cid:15) M ( k ) and the other bands. The n D Dirac pointscan appear above the M -th band, i.e. , between (cid:15) M ( k )and (cid:15) M +1 ( k ), or below the first band, i.e. , between (cid:15) ( k )and (cid:15) − ( k ). According to the I symmetry, half of the BZ( B ) must have n D / B is notunique. An example is shown in Fig. 4.) With the C z T symmetry, the number of Dirac points in B is related tothe Berry’s phase surrounding B as [130] n D π ‰ ∂ B d k · A (cid:48) ( k ) mod 2 , (36)where A (cid:48) ( k ) = (cid:80) Mn =1 i (cid:104) u n ( k ) | ∂ k u n ( k ) (cid:105) is the Berry’sconnection of the first M positive bands. In presenceof the effective inversion symmetry I , the right hand sideof the above equation is determined by the I eigenvaluesas [127, 135–137]exp (cid:18) i ‰ ∂ B d k · A (cid:48) ( k ) (cid:19) = (cid:89) K M (cid:89) n =1 ξ K ,n , (37)where K indexes the four I -invariant momenta, and ξ K ,n is the I eigenvalue of the n -th positive band at the mo-mentum K . As discussed in the last paragraph, each dou-blet at an I -invariant momentum has opposite I eigen-values. Thus, there are equal number of I eigenvalues1 and − I -invariant momenta; since the totalnumber of states at the four I -invariant momenta is 4 M ,there are 2 M eigenvalues with I = +1 and 2 M eigenval-ues with I = −
1. Hence the right hand side of Eq. (37)is 1 and we have (cid:23) ∂ B d k · A (cid:48) ( k ) = 0 mod 2 π . Accord-ing to Eq. (36), the total number of Dirac points is amultiple of 4, i.e. , n D = 0 mod 4. As we have proved inSection IV, there must be 4 l (cid:48) +2 ( l (cid:48) ∈ N ) Dirac points be-tween (cid:15) ( k ) and (cid:15) − ( k ), then the number of Dirac pointsbetween (cid:15) M ( k ) and (cid:15) M +1 ( k ) is n D − l (cid:48) − l + 2 with l = n D / − l (cid:48) − ∈ N . Thus the M -th positive band is al-ways connected to the ( M + 1)-th positive band through4 l + 2 Dirac points. According to the particle-hole sym-metry P , the M -th negative band is also connected to the( M + 1)-th negative band through 4 l + 2 Dirac points.Therefore, the whole set of bands in the system will beconnected.In general, the 4 l + 2 Dirac points between the M -thband the ( M +1)-th band can be located anywhere in theBZ. However, with the C z and the C x symmetries ofTBG, at least some of the 4 l + 2 Dirac points must locate at high symmetry point or along high symmetry lines ofthe BZ. We prove this statement by contradiction. Theunitary point group of TBG is generated by C z , C x , andthe effective inversion I and hence is isomorphic to thepoint group D d , which has 12 elements in total. If all theDirac points between the M -th band the ( M +1)-th bandare located at generic momenta, then the number of Diracpoints would be a multiple of 12, as represented by theblue dots in Fig. 5b, leading to a contradictory with the4 l + 2 Dirac points. Therefore, there must be 2 (modulo4) Dirac points at the high symmetry points or alongthe high symmetry lines. As a consequence, the entireset of bands of TBG in the second chiral limit must beconnected along the high symmetry lines. For example,as shown in Fig. 5a, there is a crossing between the 2ndand 3rd bands in the high symmetry line Γ M − K M . (Thiscrossing is protected by the effective mirror symmetry M x = C x I .) Under the actions of C x and C z , thereare in total six symmetry counterparts of this crossingpoint (including itself). Thus the number of Dirac pointsis consistent with 4 l + 2 with l = 1. VII. Conclusions
In this work, we showed that even the simple, well stud-ied BM TBG model still has several surprises related tothe deep physics that it describes. We have proved thatthe band structure in a single graphene valley of TBGis anomalous, i.e. , does not have lattice support that re-spects the C z T and P symmetries. The anomaly man-ifests as (i) a Z nontrivial topology protected P of the2 M bands (cid:15) − M ( k ) · · · (cid:15) M ( k ) for arbitrary M , providedthat the 2 M bands are gapped from other bands, (ii)4 l + 2 ( l ∈ N ) Dirac points between (cid:15) − ( k ) and (cid:15) ( k ).In the second chiral limit ( w = 0), the anomaly man-ifests as (iii) a perfect metal phase where all the bandsare connected.As a consequence of the symmetry anomaly, a faithfuldescription of TBG that respects all the symmetries ofTBG, including P , is forced to adopt a momentum spaceformalism. Any tight-binding description [1, 2, 23, 57, 66]of TBG with finite number of orbitals must break at leastone of the C z T and P symmetries (or the valley sym-metry if the tight-binding model mix the two graphenevalleys of TBG). In the other works of our series on TBG[109–113], the interacting physics is studied using a mo-mentum space formalism. Acknowledgments
We thank Aditya Cowsik and Fang Xie for valuablediscussions. This work was supported by the DOEGrant No. DE-SC0016239, the Schmidt Fund for Inno-vative Research, Simons Investigator Grant No. 404513,the Packard Foundation, the Gordon and Betty MooreFoundation through Grant No. GBMF8685 towards the1Princeton theory program, and a Guggenheim Fellowshipfrom the John Simon Guggenheim Memorial Foundation.Further support was provided by the NSF-EAGER No.DMR 1643312, NSF-MRSEC No. DMR-1420541 andDMR-2011750, ONR No. N00014-20-1-2303, Gordon and Betty Moore Foundation through Grant GBMF8685 to-wards the Princeton theory program, BSF Israel US foun-dation No. 2018226, and the Princeton Global NetworkFunds. [1] Jian Kang and Oskar Vafek. Symmetry, Maximally Lo-calized Wannier States, and a Low-Energy Model forTwisted Bilayer Graphene Narrow Bands.
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Phys. Rev. B ,85:165120, Apr 2012. A. Hamiltonian of twisted bilayer graphene inmomentum space1. The Hamiltonian
Here we briefly introduce the momentum space Hamil-tonian H ( k ) corresponding to Eq. (1). Readers may re-fer to the supplementary materials of Ref. [58] for moredetails. The basis of H ( k ) is | φ Q ,α ( k ) (cid:105) , where k is amomentum in the moir´e BZ, Q is a point in the latticeshown in Fig. 6, and α = 1 , Q lattices: the bluelattice Q T and the red lattice Q B . For Q ∈ Q T , thebasis | φ Q ,α ( k ) (cid:105) is a plane-wave state from the top layer | φ Q ,α ( k ) (cid:105) = 1 √ N (cid:88) R e i ( R + t α ) · ( k − Q ) | R + t α (cid:105) , (A1)where N is the number of lattices in the top layergraphene, R indexes all the lattices of the top layergraphene, t α is the sublattice vector of the top layergraphene, | R + t α (cid:105) is the atomic orbital at R + t α . For Q ∈ Q B , the basis | φ Q ,α ( k ) (cid:105) is a plane-wave state fromthe bottom layer | φ Q ,α ( k ) (cid:105) = 1 √ N (cid:88) R (cid:48) e i ( R (cid:48) + t (cid:48) α ) · ( k − Q ) | R (cid:48) + t (cid:48) α (cid:105) , (A2)where N is the number of lattices in the bottom layergraphene (same as the one in top layer), R (cid:48) indexes all thelattices of the bottom layer graphene, t (cid:48) α is the sublatticevector of the top layer graphene, | R (cid:48) + t (cid:48) α (cid:105) is the atomicorbital at R (cid:48) + t (cid:48) α . The Hamiltonian (Eq. (1)) on thebasis | φ Q ,α ( k ) (cid:105) is given by H Q , Q (cid:48) ( k ) = v F δ Q , Q (cid:48) ( k − Q ) · σ − θ ζ Q δ Q , Q (cid:48) ( k − Q ) × σ + (cid:88) j =1 ( δ Q (cid:48) − Q , q j + δ Q − Q (cid:48) , q j ) T j , (A3)where σ = ( σ x , σ y ) and T j ( j = 1 , ,
3) are two-by-twomatrices in the sublattice space, ζ Q = 1 for Q ∈ Q T and ζ Q = − Q ∈ Q B .
2. The periodicity of Bloch states
An eigenstate of Eq. (1) at a given momentum k canbe written as linear combination of the Bloch basis as | ψ n ( k ) (cid:105) = (cid:80) Q ,α | φ Q ,α ( k ) (cid:105) u Q α,n ( k ). Here n is the bandindex. It should be noticed that the basis states inEqs. (A1) and (A2) are not periodic in the moir´e BZsince | φ Q ,α ( k + G ) (cid:105) = | ψ Q − G ,α ( k ) (cid:105) by definition. In or-der for the Bloch state | ψ n ( k ) (cid:105) to be periodic in the moir´eBZ, i.e. , | ψ n ( k + G ) (cid:105) = | ψ n ( k ) (cid:105) , u Q ,α ( k ) should satisfy u Q ,α ( k + G ) = u Q − G ,α ( k ). We introduce the embeddingmatrix V GQ , Q (cid:48) = δ Q − G , Q (cid:48) (A4) � q q q KK' (a) q q q � M K M M M (b) FIG. 6. The Q -lattice for the momentum space Hamiltonianof twisted bilayer graphene. (a) The blue and red hexagonsrepresent the Brillouin zones of the top layer and the bottomlayer, respectively. The blue and red dots represent the posi-tions of Dirac points of the two layers in the graphene valley K , respectively. (b) The Q lattice formed by adding q , , iteratively. At each blue dot a plane-wave state from the toplayer is assigned, and at each red dot a plane-wave state fromthe bottom layer is assigned. such that we can write the periodicity of Bloch states as | u n ( k + G ) (cid:105) = V G | u n ( k ) (cid:105) , (A5)where | u n ( k ) (cid:105) = ( u Q , ,n ( k ) , u Q , ,n ( k ) , u Q , ,n ( k ) · · · ) T .While exact Bloch periodicity requires that the cutoff inthe lattice Q be large, we have showed - around the firstmagic angle - [109] that we can obtain machine precisionaccuracy in the first moir´e BZ by taking a small cutoffin Q
3. Symmetry operators in the momentum space
The crystalline symmetry group of the single val-ley Hamiltonian is the magnetic space group P (cid:48) (cid:48) C z symmetry H ( C z k ) = D ( C z ) H ( C z k ) D † ( C z ) (A6)where D Q (cid:48) , Q ( C z ) = e i π σ z δ Q (cid:48) ,C z Q . The C x symmetry H ( C x k ) = D ( C x ) H ( C x k ) D † ( C x ) (A7)where D Q (cid:48) , Q ( C x ) = σ x δ Q (cid:48) ,C x Q , and C x q = − q .The C z T symmetry H ( k ) = D ( C z T ) H ∗ Q , Q (cid:48) ( k ) D T ( C z T ) (A8)where D Q (cid:48) , Q ( C z T ) = σ x δ Q (cid:48) , Q . It should be noticedthat all rotations of momenta here are with respect tothe Γ M point of the moir´e BZ.When the second term in Eq. (A3) is negligible, H ( k )has an emergent unitary particle-hole symmetry H ( − k ) = − D ( P ) H ( k ) D † ( P ) (A9)7where D Q (cid:48) , Q ( P ) = δ Q (cid:48) , − Q ζ Q , and ζ Q = 1 for Q ∈ Q T and ζ Q = − Q ∈ Q B . The anti-unitary particle-holesymmetry P = P C z T in momentum space is H ( − k ) = − D ( P ) H ∗ ( − k ) D T ( P ) , (A10)where D Q , Q (cid:48) ( P ) = σ x δ Q (cid:48) , − Q ζ Q . When w = 0, H ( k )has an emergent chiral symmetry H ( k ) = − D † ( C ) H ( k ) D ( C ) , (A11)where D Q , Q (cid:48) ( C ) = σ z δ Q , Q (cid:48) . When w = 0, H ( k ) hasan another emergent chiral symmetry (the second chiralsymmetry) H ( k ) = − D † ( C (cid:48) ) H ( k ) D ( C (cid:48) ) , (A12)where D Q , Q (cid:48) ( C (cid:48) ) = σ z δ Q , Q (cid:48) ζ Q .The embedding matrix transform under the unitaryoperators ( g = C z , C x , P, C, C (cid:48) ) as D ( g ) V G D † ( g ) = V g G . (A13)For the unitary operator C z T , we have D ( C z T ) V G ∗ D T ( C z T ) = V G . (A14)One can verify that the two identities by explicitly actingthe symmetry operators on the embedding matrix.
4. The sewing matrices of symmetry operators
For the unitary crystalline symmetries, e.g. , g = C z , C x , P, C, C (cid:48) , we define the sewing matrices as B ( g ) n (cid:48) n ( k ) = (cid:104) u n (cid:48) ( g k ) | D ( g ) | u n ( k ) (cid:105) . (A15)For the anti-unitary symmetries, e.g. , g = C z T, P , wedefine the sewing matrices as B ( g ) n (cid:48) n ( k ) = (cid:104) u n (cid:48) ( g k ) | D ( g ) | u ∗ n ( k ) (cid:105) . (A16)Using the identities Eqs. (A13) and (A14), we obtain thatthe sewing matrices are periodic in momentum space, i.e. , B ( g ) ( k + G ) = B ( g ) ( k ) for arbitrary reciprocal lat-tice G . The explicit expression for the sewing matricesdepend on different basis, and will be give in Ref. [110]for the cases needed for our interacting problem. B. The gauge of Chern band basis